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Properties

Label 1-4008-4008.917-r0-0-0
Degree $1$
Conductor $4008$
Sign $-0.999 - 0.0403i$
Analytic cond. $18.6130$
Root an. cond. $18.6130$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.455 − 0.890i)5-s + (−0.752 + 0.658i)7-s + (0.776 + 0.629i)11-s + (−0.999 − 0.0378i)13-s + (0.0567 + 0.998i)17-s + (−0.672 + 0.739i)19-s + (0.280 + 0.959i)23-s + (−0.584 − 0.811i)25-s + (0.280 − 0.959i)29-s + (−0.169 + 0.985i)31-s + (0.243 + 0.969i)35-s + (0.553 − 0.832i)37-s + (−0.942 − 0.334i)41-s + (−0.316 + 0.948i)43-s + (−0.929 + 0.369i)47-s + ⋯
L(s)  = 1  + (0.455 − 0.890i)5-s + (−0.752 + 0.658i)7-s + (0.776 + 0.629i)11-s + (−0.999 − 0.0378i)13-s + (0.0567 + 0.998i)17-s + (−0.672 + 0.739i)19-s + (0.280 + 0.959i)23-s + (−0.584 − 0.811i)25-s + (0.280 − 0.959i)29-s + (−0.169 + 0.985i)31-s + (0.243 + 0.969i)35-s + (0.553 − 0.832i)37-s + (−0.942 − 0.334i)41-s + (−0.316 + 0.948i)43-s + (−0.929 + 0.369i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $-0.999 - 0.0403i$
Analytic conductor: \(18.6130\)
Root analytic conductor: \(18.6130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (917, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (0:\ ),\ -0.999 - 0.0403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002786089369 + 0.1380615911i\)
\(L(\frac12)\) \(\approx\) \(0.002786089369 + 0.1380615911i\)
\(L(1)\) \(\approx\) \(0.8606705983 + 0.03918073844i\)
\(L(1)\) \(\approx\) \(0.8606705983 + 0.03918073844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.455 - 0.890i)T \)
7 \( 1 + (-0.752 + 0.658i)T \)
11 \( 1 + (0.776 + 0.629i)T \)
13 \( 1 + (-0.999 - 0.0378i)T \)
17 \( 1 + (0.0567 + 0.998i)T \)
19 \( 1 + (-0.672 + 0.739i)T \)
23 \( 1 + (0.280 + 0.959i)T \)
29 \( 1 + (0.280 - 0.959i)T \)
31 \( 1 + (-0.169 + 0.985i)T \)
37 \( 1 + (0.553 - 0.832i)T \)
41 \( 1 + (-0.942 - 0.334i)T \)
43 \( 1 + (-0.316 + 0.948i)T \)
47 \( 1 + (-0.929 + 0.369i)T \)
53 \( 1 + (0.965 - 0.261i)T \)
59 \( 1 + (-0.0567 + 0.998i)T \)
61 \( 1 + (0.982 - 0.188i)T \)
67 \( 1 + (-0.455 - 0.890i)T \)
71 \( 1 + (-0.644 - 0.764i)T \)
73 \( 1 + (-0.954 + 0.298i)T \)
79 \( 1 + (-0.614 - 0.788i)T \)
83 \( 1 + (-0.997 + 0.0756i)T \)
89 \( 1 + (-0.822 - 0.569i)T \)
97 \( 1 + (-0.169 - 0.985i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.25032757013371350674872521428, −17.285787209644007297911382573608, −16.8658733832737269814941698445, −16.27205444560794230491168398687, −15.28648962036252022005454788302, −14.623411976940187207102833619200, −14.12247412541090659034487400484, −13.3699782060355690853050618107, −12.86345069486119305858239695128, −11.76049463804085609905786966776, −11.31129680023020074762893690244, −10.363746177244226724268166803904, −9.96891575282719984694225080324, −9.22489649604751813965761772877, −8.478075510677842317329216219270, −7.310374691344913456086332993688, −6.83864180851581032763940163242, −6.42588529765779323830072073470, −5.42827883993143484521142260112, −4.55652127552281344414449093347, −3.6950704298913861525538618304, −2.89049081002256217397395847990, −2.39707425658110867884099882537, −1.13305494561715248415655785776, −0.038170493607056898255295116281, 1.45192190072826434627784453023, 1.95715773977813098434347635660, 2.94272670914359661259497142962, 3.93351904417520748443975082336, 4.59424685015387829591912163757, 5.49495072519234557760595186672, 6.055713964485572964272624514658, 6.7858082626319923156192754190, 7.70010468275628952235010060628, 8.58977977347436231231639515167, 9.087276059563361350597536486756, 9.97395752718852840821952021773, 10.08929985814979573431659401943, 11.508700998272715721073652841054, 12.13379579654302921639493976058, 12.68127082086614196846972001194, 13.1091287461615467301122057068, 14.08002426817534921138220095548, 14.87392546841976911318631288766, 15.301445905780793369533896231925, 16.32219162299484084227320597281, 16.73817253220092661169371991303, 17.47245779484740106868155435996, 17.89757179955451388068358366604, 19.0535331488487415744733181470

Graph of the $Z$-function along the critical line